
theorem
  for X being set st for a being set st a in X holds a is cardinal number
  holds meet X is cardinal number
proof
  let X be set;
  assume A1: for a being set st a in X holds a is cardinal number;
  per cases;
  suppose X = {};
    hence thesis by SETFAM_1:def 1;
  end;
  suppose X <> {};
    then consider A being Cardinal such that
      A2: A in X & A = meet X by A1, Th15;
    thus thesis by A2;
  end;
end;
